You said that it must occur. Then it is not improbable, rather it is inevitable.

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Hmmm. I think semantics plays a role here.

Consider this: Given unlimited shuffles, a particular permutation of a deck of cards must eventually occur. But it is extremely improbable for any given shuffle. So, is the event that a particular permutation occurs a certainty, or very improbable?

In the context of this thread, I think we're talking about discrete events that might happen at any time, and that an improbable event means an event that is unlikely to occur in some human-scale timeframe (eg in a given year).

So although all events will occur given unlimited time, probable events will occur much more frequenty than improbable events.

My sister turns into a giant fruit bat and then builds a castle out of marshmellow. Then Barack Obama comes to visit and eats the whole thing, while an orchestra plays through the entire score of Wagner's Ring cycle. Then the Earth explodes and everybody dies except me because I happen to be rescued at the last minute by a green pixie named Horace.

Bugger. I was going to suggest us figuring out the meaning of the universe only to have it disappear instantly and be replaced by something even more unfathomable.
But I've got to hand it to you I'm afraid.

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Or me admitting I'm wrong. Maybe if Sam and I both admitted we were wrong even some of the time. But then if Tiassa and Hype did the same, well, we'd risk whales and flowers dropping out of the sky and everyone being turned into puppets.

Throw a coin so that it randomly lands somewhere on a meter stick. What are the odds that the exact center of the coin landed where it did? The odds would appear to be 1 over the number of real numbers between 0 and 100. Since there are infinitely many real numbers between 0 and 100, it would seem that the odds of the coin landing in that precise spot were zero. And yet there it is...

Consider this: Given unlimited shuffles, a particular permutation of a deck of cards must eventually occur. But it is extremely improbable for any given shuffle. So, is the event that a particular permutation occurs a certainty, or very improbable?

In the context of this thread, I think we're talking about discrete events that might happen at any time, and that an improbable event means an event that is unlikely to occur in some human-scale timeframe (eg in a given year).

So although all events will occur given unlimited time, probable events will occur much more frequenty than improbable events.

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Hmm perhaps. I didn't get the impression that the OP meant it in that way though.

Throw a coin so that it randomly lands somewhere on a meter stick. What are the odds that the exact center of the coin landed where it did? The odds would appear to be 1 over the number of real numbers between 0 and 100. Since there are infinitely many real numbers between 0 and 100, it would seem that the odds of the coin landing in that precise spot were zero. And yet there it is...

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Well, you can't measure where the 'exact' center of the coin is anymore than you can determine that my height is transcendental

Throw a coin so that it randomly lands somewhere on a meter stick. What are the odds that the exact center of the coin landed where it did? The odds would appear to be 1 over the number of real numbers between 0 and 100. Since there are infinitely many real numbers between 0 and 100, it would seem that the odds of the coin landing in that precise spot were zero. And yet there it is...

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The probability that the coin did land exactly where it did land is exactly one. That probability that it will once again land exactly where it did is exactly zero. Neither of these is constitutes "something so unlikely to happen that the odds are a multitude of multitude of billions to 1?"

The probability that the coin did land exactly where it did land is exactly one. That probability that it will once again land exactly where it did is exactly zero.

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But prior to my throw, what were the odds of the coin landing at that particular point rather than any of the other possible points along my meter stick? Were the odds exactly zero for every possible landing point? It seems to me that they were. And yet the odds of the coin landing somewhere is one (since it's stated as a premise of the problem). It seems to me that when I throw the coin it is sure to land somewhere, and there are infinitely many places it could land, but for each of the potential landing points the odds of it landing there is zero.

Surely the most improbable possibility is all the possible improbable events that could ever happen, happening at the exact same time.

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Best answer.

The most improbable event that could possibly occur, given an infinite amount of time, would be the simultaneous redistribution of all the energy in the universe to the least likely but still probabilistically possible location.

Such a thing is so monumentally improbable that you could literally live for an infinite amount of time in an infinite number of universes and still feel completely certain that you would never experience it.

Even more improbable than this would be Stephen Conroy, Minister for Broadband, Communications and the Digital Economy in Australia, growing a brain. But I hope I'm somehow almost infinitely wrong.